I have always been fascinated with this Character in the Space Marine codex. I started playing 40k at the tail end of 3rd and really got into the game at the start of 4th edition. Captain Shrike was a cool character in the 4th edition space marine codex but really didn't come into is own until the 5th edition space marine codex.
Shrike of course replaces combat tactics with the Fleet Special rule. Fleeting Space Marines are no joke and can certainly give your opponent something to think about when he fights your army. Shrike also allows a unit he is joined with at the beginning of the game to have the infiltrate rule. With that you can take one unit and either infiltrate or outflank. Shrike overall is pretty fragile. He only has a 3+ save and a 4+ inv save with 3 wounds. His lightning claws do rend which is nice but other than that he himself doesn't have very many bells and whistles.
When you first read about Shrike you kind of go "wow this guy would be awesome with a squad of vanguard vets that infiltrate or outflank. That means a first turn charge and some serious damage." That is where my thinking first went to but after really looking at the advantages Shrike gives I was disinclined to think that way because of the cost of the unit and how fragile it really was. My next thought was why not get buy 8 TH/SS terminators and stick them in a LRC and charge across the field and maybe take a command squad as support for them. This is an ok approach but I don't think that it is the best way. Why? Because of so many people taking lance and melta spam that in my opinion the LR and LRC are becoming obsolete on the 40k battle field and most people are leaning towards lighter armour vehicles.
So what is the best way to run Shrike in the current set of rules? It all comes down to what your are fielding in your Space Marine army. Recently, I have been running Shrike with 10 TH/SS terminators. Of course I don't run them like most people and try to infiltrate them every game so that they can get the shit shot out of them. I use them in different battle field roles depending on what army I am playing against. Sometimes I outflank with them and sometimes I just line them up in front of my battle line. It all depends on what I am facing.
Overall I think that Shrike is an underutilized character in the Space Marine codex that people ignore in favor of Lysander, Vulkan, and Khan. I feel that in the right army build and players hands that what he brings to the table will certainly be a force to be reckoned with.